Lemma 15.102.2. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $N$ be an $A$-module annihilated by $I$. There exists an integer $n > 0$ such that $\text{Tor}^ A_ p(I^ nM, N) \to \text{Tor}^ A_ p(M, N)$ is zero for all $p \geq 0$.
Proof. By Lemma 15.101.7 we can factor $I^ nM \to M$ as $I^ nM \to M \otimes _ A^\mathbf {L} I \to M$. We claim the composition
\[ I^ nM \otimes _ A^\mathbf {L} N \to (M \otimes _ A^\mathbf {L} I) \otimes _ A^\mathbf {L} N \to M \otimes _ A^\mathbf {L} N \]
is zero. Namely, the diagram
\[ \xymatrix{ (M \otimes _ A^\mathbf {L} I) \otimes _ A^\mathbf {L} N \ar[rr] \ar[rd] & & M \otimes _ A^\mathbf {L} (I \otimes _ A^\mathbf {L} N) \ar[ld] \\ & M \otimes _ A^\mathbf {L} N } \]
commutes (details omitted) and the map $I \otimes _ A^\mathbf {L} N \to N$ is zero as $N$ is annihilated by $I$. $\square$
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